Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(668\!\cdots\!569\)\(\medspace = 23^{10} \cdot 709^{12} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.5.11561663.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.5.11561663.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - 3x^{5} + 5x^{4} + 2x^{3} - 8x^{2} - 2x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 311 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 311 }$:
\( x^{2} + 310x + 17 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 247 a + 33 + \left(16 a + 221\right)\cdot 311 + \left(23 a + 182\right)\cdot 311^{2} + \left(118 a + 249\right)\cdot 311^{3} + \left(206 a + 242\right)\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 256\cdot 311 + 198\cdot 311^{2} + 92\cdot 311^{3} + 86\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 105 + 264\cdot 311 + 16\cdot 311^{2} + 213\cdot 311^{3} + 152\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 + 126\cdot 311 + 126\cdot 311^{2} + 251\cdot 311^{3} + 32\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 259 a + 268 + \left(294 a + 174\right)\cdot 311 + \left(22 a + 245\right)\cdot 311^{2} + \left(90 a + 12\right)\cdot 311^{3} + \left(173 a + 2\right)\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 64 a + 280 + \left(294 a + 301\right)\cdot 311 + \left(287 a + 188\right)\cdot 311^{2} + \left(192 a + 33\right)\cdot 311^{3} + \left(104 a + 20\right)\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 52 a + 216 + \left(16 a + 210\right)\cdot 311 + \left(288 a + 284\right)\cdot 311^{2} + \left(220 a + 79\right)\cdot 311^{3} + \left(137 a + 85\right)\cdot 311^{4} +O(311^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character value |
| $1$ | $1$ | $()$ | $20$ |
| $21$ | $2$ | $(1,2)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.